The strict minimum message length (SMML) principle links data compression with inductive inference. The corresponding estimators have many useful properties but they can be hard to calculate. We investigate SMML estimators for linear regression models and we show that they have close connections to hyperbolic geometry. When equipped with the Fisher information metric, the linear regression model with $p$ covariates and a sample size of $n$ becomes a Riemannian manifold, and we show that this is isometric to $(p+1)$-dimensional hyperbolic space $\mathbb{H}^{p+1}$ equipped with a metric tensor which is $2n$ times the usual metric tensor on $\mathbb{H}^{p+1}$. A natural identification then allows us to also view the set of sufficient statistics for the linear regression model as a hyperbolic space. We show that the partition of an SMML estimator corresponds to a tessellation of this hyperbolic space.